Number Systems

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# Number Systems - PowerPoint PPT Presentation

Number Systems. Computing Theory – F453. Data Representation. Data in a computer needs to be represented in a format the computer understands. This does not necessarily mean that this format is easy for us to understand. Not easy, but not impossible!

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Number Systems

Computing Theory – F453

Data Representation
• Data in a computer needs to be represented in a format the computer understands.
• This does not necessarily mean that this format is easy for us to understand.
• Not easy, but not impossible!
• A computer only understand the concept of ON and OFF.
• Why?
• How do we translate this into something WE understand?
• We use a numeric representation (1s and 0s)
Data Representation
• If a computer can only understand ON and OFF, which is represented by 1 and 0, then which is which?
• 1 = ON
• 0 = OFF
• This is known as the Binary system.
• Because there are only 2 digits involved, it is known as Base 2.
• But what does it MEAN??!
Denary Numbers
• We use the Denary Number System.
• This is in Base 10, because there are 10 single digits in our number system.
• Why? We are surrounded by things that are divisible by ten.
• Counting in tens is not a new phenomenon…
• Even the Egyptians did it!
Real Numbers
• In the real world we have to work with decimal numbers, but there is no place in binary for a decimal point. In these cases, we need to ‘normalise’ the number.
• In short, we place everything to the right of the decimal point:

Positive exponent represents the decimal point moving left

10111.0

0.10111 x 25

0.10111 x 2101

010111101

Exponent

Mantissa

Real Numbers
• Example two:
• The binary number 10.11011 with 8 bits for the mantissa and 6 for the exponent

.1011011

0. .1011011 x 22

0.10111 x 210

01011011000010

Positive exponent represents the decimal point moving left

Exponent

Mantissa

Real Numbers
• Example two:
• The denary number 37.5 with 8 bits for the mantissa and 6 for the exponent

37.5 =

100101.1

0.1001011 x 26

0.1001011 x 2110

01001011000110

Positive exponent represents the decimal point moving left

Exponent

Mantissa

Real Numbers – Your Turn
• Example three:
• The denary number 52.75 with 8 bits for the mantissa and 8 for the exponent

52.75 =

110100.11

0.11010011 x 26

0.11010011 x 2110

01101001100000110

Positive exponent represents the decimal point moving left

Exponent

Mantissa

Real Numbers – Your Turn
• Example four:
• The denary number 22.8125 with 10 bits for the mantissa and 5 for the exponent

22.8125 =

010110.1101

0.101101101 x 25

0.101101101 x 2101

010110110101101

Positive exponent represents the decimal point moving left

Exponent

Mantissa

Real Numbers – In reverse
• Example:
• The binary number 01011100000011 with 8 bits for the mantissa and 6 for the exponent

Mantissa

Exponent

01011100000011

0.1011100 x 211

0.1011100 x 23

0101.1100

= 5.75

Real Small Numbers
• In the real world we also have to work with numbers which are less than 1, or decimals.
• This is tackled in the same way, but we make use of two’s complement for the exponent:

Negative exponent represents the decimal point moving right

0.00010101

0.10101 x 2-3

0.10101 x 2-11

0.10101 x 2-11

01010111110101

Mantissa

Exponent

Real Numbers – Your Turn
• Example five:
• The binary number 0.00110 with 8 bits for the mantissa and 8 for the exponent

0.00110

0.110 x 2-2

0.110 x 2-10

0.110 x 211111110

0000011011111110

Exponent

Mantissa

Real Numbers – In reverse
• Example:
• The binary number 01011110111010 with 8 bits for the mantissa and 6 for the exponent

Mantissa

Exponent

01011110111010

0.1011110 x 2-110

0.1011110 x 2-6

0.0000001011110

Normalisation
• In the examples, the point in the mantissa is always placed before the first zero (eg. 0.11010).
• This not only allows for the maximum number to be held, but by ensuring that the first two digits are different, the mantissa is said to be normalised.
• Therefore, for a positive number, the first digit is always a zero, and the exponent is held in two’s complement form.